Editing Comparative statics (section)

====An example of the role of the stability assumption====
Suppose that the quantities demanded and supplied of a product are determined by the following equations:

:<math>Q^{d}(P) = a + bP</math>

:<math>Q^{s}(P) = c + gP</math>

where <math>Q^{d}</math> is the quantity demanded, <math>Q^{s}</math> is the quantity supplied, ''P'' is the price, ''a'' and ''c'' are intercept parameters determined by exogenous influences on demand and supply respectively, ''b'' < 0 is the reciprocal of the slope of the [[demand curve]], and ''g'' is the reciprocal of the slope of the supply curve; ''g'' > 0 if the supply curve is upward sloped, ''g'' = 0 if the supply curve is vertical, and ''g'' < 0 if the supply curve is backward-bending. If we equate quantity supplied with quantity demanded to find the equilibrium price <math>P^{eqb}</math>, we find that

:<math>P^{eqb}=\frac{a-c}{g-b}.</math>

This means that the equilibrium price depends positively on the demand intercept if ''g'' – ''b'' > 0, but depends negatively on it if ''g'' – ''b'' < 0.  Which of these possibilities is relevant? In fact, starting from an initial static equilibrium and then changing ''a'', the new equilibrium is relevant ''only'' if the market actually goes to that new equilibrium.  Suppose that price adjustments in the market occur according to

:<math>\frac{dP}{dt}=\lambda (Q^{d}(P) - Q^{s}(P))</math>

where <math>\lambda</math> > 0 is the speed of adjustment parameter and <math>\frac{dP}{dt}</math> is the [[time derivative]] of the price — that is, it denotes how fast and in what direction the price changes.  By [[stability theory]], ''P'' will converge to its equilibrium value if and only if the [[derivative]] <math>\frac{d(dP/dt)}{dP}</math> is negative.  This derivative is given by

:<math> \frac{d(dP/dt)}{dP} = - \lambda(-b+g).</math>

This is negative if and only if ''g'' – ''b'' > 0, in which case the demand intercept parameter ''a'' positively influences the price.  So we can say that while the direction of effect of the demand intercept on the equilibrium price is ambiguous when all we know is that the reciprocal of the supply curve's slope, ''g'', is negative, in the only relevant case (in which the price actually goes to its new equilibrium value) an increase in the demand intercept increases the price.  Note that this case, with ''g'' – ''b'' > 0, is the case in which the supply curve, if negatively sloped, is steeper than the demand curve.